CCF - Compare and Contrast Functions (Overview)
Compare and Contrast Functions
Introduction
Essential Questions
- How do I use graphs to represent and solve real-world equations and inequalities?
- Why is the concept of a function important, and how do I use function notation to show a variety of situations modeled by functions?
- How do I interpret functions that arise in applications in terms of context?
- How do I use different representations to analyze linear and exponential functions?
- How do I build a linear or exponential function that models a relationship between two quantities?
- How do I build new functions from existing functions?
- How can we use real-world situations to construct and compare linear and exponential models and solve problems?
- How do I interpret expressions for functions in terms of the situation they model?
- How is a relation determined to be linear, quadratic, or exponential?
- What are the specific features that distinguish the graphs of linear, quadratic, and exponential functions from one another?
Key Terms
The following key terms will help you understand the content in this module.
Arithmetic Sequence - A sequence of numbers in which the difference between any two consecutive terms is the same.
Average Rate of Change - The change in the value of a quantity by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.
Coefficient - A number multiplied by a variable in an algebraic expression.
Constant Rate of Change - With respect to the variable x of a linear function y = f(x), the constant rate of change is the slope of its graph.
Continuous - Describes a connected set of numbers, such as an interval.
Discrete - A set with elements that are disconnected.
Domain - The set of x-coordinates of the set of points on a graph. The set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
End Behaviors - The appearance of a graph as it is followed farther and farther in either direction.
Explicit Expression - A formula that allows direct computation of any term for a sequence a 1 , a 2 , a 3 , . . . , a n , . . . .
Exponential Function - A nonlinear function in which the independent value is an exponent in the function, as in y = ab x.
Exponential Model - An exponential function representing real-world phenomena. The model also represents patterns found in graphs and/or data.
Expression - Any mathematical calculation or formula combining numbers and/or variables using sums, differences, products, quotients including fractions, exponents, roots, logarithms, functions, or other mathematical operations.
Even Function - A function with a graph that is symmetric with respect to the
Factor - For any number x, the numbers that can be evenly divided into x are called factors of x.
Geometric Sequence - A sequence of numbers in which the ratio between any two consecutive terms is the same. In other words, you multiply by the same number each time to get the next term in the sequence. This fixed number is called the common ratio for the sequence.
Horizontal Shift - A rigid transformation of a graph in a horizontal direction, either left or right.
Interval Notation - A notation representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included.
Linear Function - A function with a constant rate of change and a straight line graph.
Linear Model - A linear function representing real-world phenomena. The model also represents patterns found in graphs and/or data.
Odd Function - A function with a graph that is symmetric with respect to the origin. A function is odd if and only if
Parameter - The independent variable or variables in a system of equations with more than one dependent variable.
Quadratic Equation - An equation of degree 2, which has at most two solutions.
Quadratic Function - A function of degree 2 which has a graph that "turns around" once, resembling an umbrella-like curve that faces either right-side up or upside down. This graph is called a parabola.
Root - The x -values where the function has a value of zero.
Range - The set of y-coordinates of the set of points on a graph. The set of y-coordinates of a given set of ordered pairs. The set of all possible outputs of a function or relation.
Recursive Formula - A formula that requires the computation of all previous terms to find the value of an .
Slope - The ratio of the vertical and horizontal changes between two points on a surface or a line.
Term - A value in a sequence--the first value in a sequence is the 1st term, the second value is the 2nd term, and so on. A term is also any of the monomials that make up a polynomial.
Vertical Translation - A shift in which a plane figure moves vertically.
x-intercept - The point where a line meets or crosses the x-axis.
y-intercept - The point where a line meets or crosses the y-axis.
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